Green's Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space
We consider the Friedrichs self-adjoint extension for a differential operator A of the form A=A0+q(x)⋅, which is defined on a bounded domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of o...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/902638 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We consider the Friedrichs self-adjoint extension for a differential
operator A of the form A=A0+q(x)⋅, which is defined on a bounded
domain Ω⊂ℝn, n≥1 (for n=1 we assume that Ω=(a,b) is a finite
interval). Here A0=A0(x,D) is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth
coefficients and a potential q(x) is a real-valued integrable function
satisfying the generalized Kato condition. Under these assumptions
for the coefficients of A and for positive λ large enough we obtain the
existence of Green's function for the operator A+λI and its estimates
up to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω of Fourier
series in eigenfunctions of this operator. In particular, these results
can be applied for the basis of the Fourier method which is usually
used in practice for solving some equations of mathematical physics. |
|---|---|
| ISSN: | 1085-3375 1687-0409 |